Season 1, episode 13.
Original air date: 5 June 1952.
Friday and Jacobs are called to the apartment of a man who phoned in the fatal shooting of his wife. He claims it was suicide, but upon looking at the crime scene and talking to neighbors they begin to wonder if it was murder.
Cast: Jack Webb (Sgt. Joe Friday), Barney Phillips (Sgt. Ed Jacobs), Harry Bartell (Andrew Robertson), Herb Butterfield (Lt. Lee Jones), Jack Kruschen (Ted Carleton), Ken Peters (Lt. Harry Fremont), Will Yeager (George Wright), George Finneman (Announcer opening: uncredited) and Hal Gidney (Announcer closing: uncredited).
From IMDB.

This is the kind of story that I really enjoy Dragnet for. A straight open-and-shut case, all signs point to murder. Friday and Jacobs (his partner) lean on the suspect to admit his crime. He steadily denies everything. His word against the police.

Suddenly Lieut. Lee Jones, of the Crime Lab, finishes his off screen investigation. All the evidence, previously so incriminating, changes like a kaleidoscope image. Not only does it change to look like a suicide (As he claimed), but murder is shown to be impossible. Exoneration.

Definitely worth seeing to compare with the Sherlock Holmes-types who are never wrong. They got it wrong here and took it in stride.